Finite Group Databasebeta

← #89970

Subgroups — #89970

Order 448.

168 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#6553C211yesmaximal, characteristic, complemented
2C7×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
3#6503C211yesmaximal, characteristic, complemented
4#6495C211yesmaximal, characteristic, complemented
5#6411C211yesmaximal, characteristic, complemented
6#6399C211yesmaximal, characteristic, complemented
7#6397C211yesmaximal, characteristic, complemented
8C7×#41C2211yescharacteristic
9C2×C7×D4C2211yescharacteristic, complemented
10C2×C22⋊D7—22no—
11#1161—21no—
12D4×D7—22no—
13D4×D7—21no—
14#1157—21no—
15#1157C2211yescharacteristic
16C2×D28—21no—
17C2×D28C2211yescharacteristic
18C7×SD16—21no—
19C7×D8—21no—
20C7×M16C2211yescharacteristic, complemented
21#1142—21no—
22#1141—21no—
23C4×Dic7C2211yescharacteristic, complemented
24#1136C2211yescharacteristic, complemented
25#453—71nomaximal, Sylow
26C23×C7—21no—
27C22×D7—22no—
28C22×D7D412yes—
29C7×Q8D411yescharacteristic, complemented
30C7×D4—21no—
31C7×D4—21no—
32C7×D4—21no—
33C7×D4D411yescharacteristic, complemented
34C2×C4×C7—21no—
35C2×C4×C7C2311yescharacteristic
36C22⋊D7—22no—
37C22⋊D7—21no—
38C22⋊D7—44no—
39C2×Dic7—21no—
40D28—22no—
41D28—21no—
42D28—21no—
43D28D411yescharacteristic, complemented
44C4×D7—22no—
45C4×D7—21no—
46Dic14D411yescharacteristic, complemented
47C7×C8—21no—
48C7⋊C8—21no—
49H(5,𝔽2)—71no—
50AGL(1,ℤ/8)—71no—
51AGL(1,ℤ/8)D711yescharacteristic, complemented
52#127—71no—
53#104—71no—
54#104—71no—
55#100—71no—
56C22×C7—21no—
57C22×C7—42no—
58C22×C7—41no—
59C22×C7C2×D411yescharacteristic
60C2×D7—22no—
61C2×D7—21no—
62C2×D7—42no—
63C2×D7—82no—
64C4×C7—21no—
65C4×C7C2×D411yescharacteristic
66C4×C7C2×D411yescharacteristic
67Dic7—22no—
68Dic7—21no—
69#41—71no—
70#41—141no—
71#41—141no—
72#41C2×D711yescharacteristic
73C2×D4—71no—
74C2×D4—141no—
75C2×D4—142no—
76C2×D4—141no—
77C2×D4—142no—
78C2×D4C2×D711yescharacteristic, complemented
79SD16—21no—
80SD16—141no—
81D8—21no—
82D8—141no—
83M16—71no—
84M16C2×D711yescharacteristic, complemented
85C42—71no—
86C2×C7—21no—
87C2×C7—41no—
88C2×C7—81no—
89C2×C7#12011yescharacteristic
90D7—42no—
91D7—41no—
92C23—21no—
93C23—72no—
94C23—142no—
95Q8—71no—
96Q8C22⋊D711yescharacteristic, complemented
97D4—21no—
98D4—21no—
99D4—21no—
100D4—71no—
101D4—142no—
102D4—142no—
103D4—141no—
104D4—141no—
105D4—141no—
106D4—284no—
107D4C22⋊D711yescharacteristic, complemented
108C2×C4—21no—
109C2×C4—142no—
110C2×C4—141no—
111C2×C4—141no—
112C2×C4C22×D711yescharacteristic
113C8—21no—
114C8—141no—
115C7#45311yescharacteristic, complemented, Sylow
116C22—21no—
117C22—42no—
118C22—41no—
119C22—141no—
120C22—142no—
121C22—282no—
122C22—562no—
123C22D4×D711yescharacteristic
124C4—21no—
125C4—142no—
126C4—141no—
127C4D4×D711yescharacteristic
128C4C2×C22⋊D711yescharacteristic
129C2—21no—
130C2—41no—
131C2—81no—
132C2—282no—
133C2—281no—
134C2#650011yescharacteristic
subgroupG/H
centerC2#6500
commutator, cosocleC2×C4×C7C23
FrattiniC2×C4C22×D7
FittingC7×AGL(1,ℤ/8)C2
socleC2×C7#120
hypercenterAGL(1,ℤ/8)D7
hypocenterC7#453
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #6553C2semidirect
2. C7×AGL(1,ℤ/8)C2semidirect
3. #6503C2semidirect
4. #6495C2semidirect
5. #6411C2semidirect
6. #6399C2semidirect
7. #6397C2semidirect
9. C2×C7×D4C22semidirect
20. C7×M16C22semidirect
23. C4×Dic7C22semidirect
24. #1136C22semidirect
29. C7×Q8D4semidirect
33. C7×D4D4semidirect
43. D28D4semidirect
46. Dic14D4semidirect
51. AGL(1,ℤ/8)D7semidirect
78. C2×D4C2×D7semidirect
84. M16C2×D7semidirect
96. Q8C22⋊D7semidirect
107. D4C22⋊D7semidirect
115. C7#453semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#65532
C7×AGL(1,ℤ/8)2
#65032
#64952
#64112
#63992
#63972
#4537